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Properties

Label 16.12.9081910848...7168.1
Degree $16$
Signature $[12, 2]$
Discriminant $2^{56}\cdot 3^{8}\cdot 577^{3}$
Root discriminant $64.55$
Ramified primes $2, 3, 577$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 16T1379

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-162479, 1011312, -824504, -1937600, -176464, 654904, 289256, -17568, -61060, -18160, 2544, 2400, 392, -88, -40, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 40*x^14 - 88*x^13 + 392*x^12 + 2400*x^11 + 2544*x^10 - 18160*x^9 - 61060*x^8 - 17568*x^7 + 289256*x^6 + 654904*x^5 - 176464*x^4 - 1937600*x^3 - 824504*x^2 + 1011312*x - 162479)
 
gp: K = bnfinit(x^16 - 40*x^14 - 88*x^13 + 392*x^12 + 2400*x^11 + 2544*x^10 - 18160*x^9 - 61060*x^8 - 17568*x^7 + 289256*x^6 + 654904*x^5 - 176464*x^4 - 1937600*x^3 - 824504*x^2 + 1011312*x - 162479, 1)
 

Normalized defining polynomial

\( x^{16} - 40 x^{14} - 88 x^{13} + 392 x^{12} + 2400 x^{11} + 2544 x^{10} - 18160 x^{9} - 61060 x^{8} - 17568 x^{7} + 289256 x^{6} + 654904 x^{5} - 176464 x^{4} - 1937600 x^{3} - 824504 x^{2} + 1011312 x - 162479 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[12, 2]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(90819108489560418567132807168=2^{56}\cdot 3^{8}\cdot 577^{3}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $64.55$
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
Ramified primes:  $2, 3, 577$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is not a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $\frac{1}{47} a^{13} + \frac{7}{47} a^{12} + \frac{5}{47} a^{11} - \frac{16}{47} a^{10} + \frac{10}{47} a^{9} - \frac{8}{47} a^{8} - \frac{2}{47} a^{7} - \frac{8}{47} a^{6} - \frac{11}{47} a^{5} + \frac{23}{47} a^{4} - \frac{15}{47} a^{3} - \frac{13}{47} a^{2} - \frac{19}{47} a$, $\frac{1}{235} a^{14} - \frac{2}{235} a^{13} - \frac{21}{47} a^{12} + \frac{16}{47} a^{11} + \frac{107}{235} a^{10} - \frac{4}{235} a^{9} - \frac{71}{235} a^{8} - \frac{37}{235} a^{7} + \frac{108}{235} a^{6} - \frac{66}{235} a^{5} - \frac{34}{235} a^{4} - \frac{19}{235} a^{3} - \frac{18}{47} a^{2} - \frac{64}{235} a + \frac{2}{5}$, $\frac{1}{62084277421105219918595921727751267765} a^{15} + \frac{72550054706402882159208392313546807}{62084277421105219918595921727751267765} a^{14} - \frac{421147911774527817134879300079839968}{62084277421105219918595921727751267765} a^{13} - \frac{5817211584928147972426196841448925112}{12416855484221043983719184345550253553} a^{12} + \frac{26632065160307069107456434956129937432}{62084277421105219918595921727751267765} a^{11} - \frac{10483128937588772697313660233762456236}{62084277421105219918595921727751267765} a^{10} - \frac{20079276536668243055980100946791303192}{62084277421105219918595921727751267765} a^{9} + \frac{5223003079181377294366158312805432544}{62084277421105219918595921727751267765} a^{8} + \frac{3829377936313850213303072399800504327}{12416855484221043983719184345550253553} a^{7} - \frac{1332565443334071296690811386483544044}{62084277421105219918595921727751267765} a^{6} + \frac{30933932890819755203256180692011395097}{62084277421105219918595921727751267765} a^{5} - \frac{446811059199952960942033850033445794}{12416855484221043983719184345550253553} a^{4} + \frac{15122494819213126651710402857431257284}{62084277421105219918595921727751267765} a^{3} - \frac{4197485387529344594300153465413060289}{62084277421105219918595921727751267765} a^{2} + \frac{5379640126012965667295680490121783833}{62084277421105219918595921727751267765} a + \frac{136591815220002017183594995197594193}{1320942072789472764225445143143643995}$

magma: IntegralBasis(K);
 
sage: K.integral_basis()
 
gp: K.zk
 

Class group and class number

Trivial group, which has order $1$ (assuming GRH)

magma: ClassGroup(K);
 
sage: K.class_group().invariants()
 
gp: K.clgp
 

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $13$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( -1 \) (order $2$)
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
Fundamental units:  Units are too long to display, but can be downloaded with other data for this field from 'Stored data to gp' link to the right (assuming GRH)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 1200446045.56 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1379:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 2048
The 71 conjugacy class representatives for t16n1379 are not computed
Character table for t16n1379 is not computed

Intermediate fields

\(\Q(\sqrt{2}) \), \(\Q(\sqrt{3}) \), \(\Q(\sqrt{6}) \), 4.4.18432.1, \(\Q(\zeta_{16})^+\), \(\Q(\sqrt{2}, \sqrt{3})\), \(\Q(\zeta_{48})^+\)

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Degree 16 siblings: data not computed
Degree 32 siblings: data not computed

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type R R ${\href{/LocalNumberField/5.8.0.1}{8} }{,}\,{\href{/LocalNumberField/5.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.8.0.1}{8} }{,}\,{\href{/LocalNumberField/13.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/29.8.0.1}{8} }{,}\,{\href{/LocalNumberField/29.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/37.8.0.1}{8} }{,}\,{\href{/LocalNumberField/37.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/43.8.0.1}{8} }{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{10}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

magma: p := 7; // to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
magma: idealfactors := Factorization(p*Integers(K)); // get the data
 
magma: [<primefactor[2], Valuation(Norm(primefactor[1]), p)> : primefactor in idealfactors];
 
sage: p = 7; # to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
sage: [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
 
gp: p = 7; \\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
gp: idealfactors = idealprimedec(K, p); \\ get the data
 
gp: vector(length(idealfactors), j, [idealfactors[j][3], idealfactors[j][4]])
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
2Data not computed
3Data not computed
577Data not computed